Aristotle's very first counterargument has the premise "Some animal is
not a human" and conclusion "Some human is not an animal" [PRIOR
ANALYTICS 25 a 10-15]. His point, of course, which he explicitly makes,
is that even though the particular affirmative "converts" - implies its
own converse, "Some animal is a human" implies "Some human is an animal"
- such is not the case with the closely related particular negative.
[POINT 1] Boole's 1847 MATHEMATICAL ANALYSIS OF LOGIC, pages 23, 25 and
29, takes the equation 'vxy = 0' for "Some Xs are not Ys". In light of
his laws, notably 'xy = yx', the equation 'vxy = 0' ["Some Xs are not
Ys"] yields 'vyx = 0' [or "Some Ys are not Xs"]. Thus, Boole's 1847
"translation" is inadequate or his logic is unsound. [POINT 2] I would
never have noticed this if I had not been inspired to go looking for it
by reading Peirce's brilliant 1865 Harvard Lecture "Boole's Calculus of
Logic" in which he points out that the equation Boole uses in the 1854
LAWS OF THOUGHT for "Some Xs are not Ys" yields the corresponding one
for "Some Ys are not Xs". Thus, although Boole made changes for the 1854
work, he did not correct this problem: Boole's 1854 "translation" is
inadequate or his logic is unsound. [POINT 3] Boole nowhere in his
writings has even one counterargument [countermodel,
counterinterpretation, independence example, or whatever your preferred
terminology may be]. [POINT 4] Boole nowhere in his writings explicitly
addresses the issue of the convertibility of the particular negative
that Aristotle settled so dramatically in PRIOR ANALYTICS 25a10-15.
Q1 Has anyone noted point 1 before in print?
Q2 Has anyone noted the Peirce criticism of Boole in point 2 before in
print?
Q3 Has anyone any explanation for point 3?
Q4 Has anyone any explanation for point 4?
Q5 Can we be sure that Boole actually read even the first two chapters
of PRIOR ANALYTICS?
-